The state of
equilibrium (where R=1) can help us explore the immunisation level required for the
eradication of an infectious agent.

From the definition of R (see previous slide),
it is:
R = R0 (S/N) [S: number of susceptibles; I: number of immunes; N: total
population].

In the state of equilibrium, it is:
R = R0 (S/N)

eq = 1 [(S/N)eq: the fraction
of susceptibles in the state of equilibrium] or
R = R0 [1 - (I/N)eq] = 1 [(I/N)eq: the fraction
of immunes in the state of equilibrium; we assume that, at equilibrium, the number of
infected cases is negligible, therefore S+I=N or S/N+I/N=1] or
R0 = 1 / [1 - (I/N)eq].

If we solve for (I/N)

eq, we get:
(I/N)eq = 1 - (1/R0).

On reflection, we realise that (I/N)

eq is the necessary
proportion of immunes in the population in order to have equilibrium (R=1). In other
words, if the proportion of immunes were higher than this, R would be smaller
than 1, and the microorganism would be eventually eradicated.

If this level of population immunity is achieved by mass vaccination, we can consider
(I/N)

eq as the critical proportion of vaccination level
above which we would eventually accomplish eradication of the microorganism (PC).
Therefore, PC = 1 - (1/R0).